Optimal diameter computation within bounded clique-width graphs

TL;DR

This paper introduces an efficient algorithm for computing all eccentricities, Wiener index, and median set in graphs of bounded clique-width, matching known lower bounds and improving label size dependencies.

Contribution

It presents a new algorithm that computes eccentricities, Wiener index, and median set in graphs of bounded clique-width within optimal time bounds, and develops a compact distance-labeling scheme.

Findings

Algorithm computes all eccentricities in O(2^{O(k)}(n+m)^{1+o(1)}) time.

Distance-labeling scheme uses O(k log^2 n) bits per vertex.

Provides an O(k n^2 log n)-time algorithm for All-Pairs Shortest Paths in graphs of bounded clique-width.

Abstract

Coudert et al. (SODA'18) proved that under the Strong Exponential-Time Hypothesis, for any $\epsilon >0$, there is no ${\cal O}(2^{o(k)}n^{2-\epsilon})$-time algorithm for computing the diameter within the $n$-vertex cubic graphs of clique-width at most $k$. We present an algorithm which given an $n$-vertex $m$-edge graph $G$ and a $k$-expression, computes all the eccentricities in ${\cal O}(2^{{\cal O}(k)}(n+m)^{1+o(1)})$ time, thus matching their conditional lower bound. It can be modified in order to compute the Wiener index and the median set of $G$ within the same amount of time. On our way, we get a distance-labeling scheme for $n$-vertex $m$-edge graphs of clique-width at most $k$, using ${\cal O}(k\log^2{n})$ bits per vertex and constructible in ${\cal O}(k(n+m)\log{n})$ time from a given $k$-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on $k$ in their scheme. As a corollary, we get an ${\cal O}(kn^2\log{n})$-time algorithm for computing All-Pairs Shortest-Paths on $n$-vertex graphs of clique-width at most $k$. This partially answers an open question of Kratsch and Nelles (STACS'20).